Question

Which function has an inverse that is also a function? G(x) = 2x – 3 k(x) = –9x2 f(x) = |x + 2| w(x) = –20

Answer 1

Answer:

The function whose inverse is also a function is:

             $G(x)=2x-3$

Step-by-step explanation:

We know that inverse of a function is also a function if it is one-to-one function.

i.e. the graph of the function satisfies the horizontal line test.

i.e. there is a unique image corresponding to each element.

1)

  $G(x)=2x-3$

The function G(x) is a straight increasing line( since the slope is positive)

Hence, the function is 1-1 as it is a strictly increasing function.

Hence, the inverse of G(x) is again a function.

2)

  $K(x)=-9x^2$

We know that a quadratic function is never 1-1.

( Since, it repeats it's value for every negative and positive value of a number )

Hence, the inverse of K(x) is not a function.

3)

     $f(x)=|x+2|$

Again we know that the modulus function is not a 1-1 function.

Hence, it's inverse is also not a function.

4)

            $w(x)=-20$

As we know that a constant function is not 1-1.

Since it takes same value for each x.

Hence, the inverse of w(x) is not a function.

Answer 2

Answer:

G(x) = 2x - 3 has a linear inverse which is a function.

k(x) = -9x2 has a square root inverse that is a function on only the interval x < 0.

Step-by-step explanation:

The inverse of a function is a reflection across the y=x line. This results in switching the values of the input and output or (x,y) points to become (y,x). This can be done algebraically in an equation as well. Begin by switching the x and y in the equation then solve for y.

x = 2y - 3 -------> y = x/2 + 3/2 is linear and a function.

-9y2 = x --------> y = √(-x/9)  is a square root that is a function on only the interval x < 0.

Absolute value does not have an inverse function.

x = -20 is a vertical line which is not a function.